# example of cohomology and Mayer-Vietoris sequence

Consider n-dimensional sphere ${S}^{n}=\{v\in {R}^{n+1}|\mathit{\hspace{1em}}|v|=1\}$.

Let $A=\{({x}_{0},\mathrm{\dots},{x}_{n})\in {S}^{n}|\mathit{\hspace{1em}}{x}_{0}>-1/2\}$ and $$.

Of course both $A$ and $B$ are open (in ${S}^{n}$) and their union is ${S}^{n}$. Furthermore, it can be easily seen, that their intersection can be contracted into ”big circle”, i.e. $A\cap B$ has homotopy type^{} of ${S}^{n-1}$. Also both $A$ and $B$ are contractible^{} (they are homeomorphic^{} to ${R}^{n}$ via stereographic projection). So, write part of a Meyer-Vietoris sequence (for the cohomology^{} ${H}^{m}(X)={H}^{m}(X,G)$, where $G$ is a fixed Abelian group^{}):

$\mathrm{\cdots}\to {H}^{m}(A)\oplus {H}^{m}(B)\to {H}^{m}(A\cap B)\to {H}^{m+1}({S}^{n})\to {H}^{m+1}(A)\oplus {H}^{m+1}(B)\to \mathrm{\cdots}$

Since both $A$ and $B$ are contractible and $A\cap B$ is homotopic^{} to ${S}^{n-1}$, we have the following short exact sequence:

$0\to {H}^{m}({S}^{n-1})\to {H}^{m+1}({S}^{n})\to 0$

which shows that ${H}^{m}({S}^{n-1})$ is isomorphic^{} to ${H}^{m}({S}^{n})$ for every $n>0$ and $m>0$. So, in order to calculate cohomology groups^{} of spheres, we only need to know the cohomology groups of ${S}^{1}$. And those can be also calculated, if we once again apply previous schema. Note, that in the case of ${S}^{1}$ we have that $A\cap B$ has the homotopy type of a discrete space with two points. Therefore all their cohomology groups are trivial, except for ${H}^{0}$ (which can be easily calculated to be equal to ${H}^{0}(*)\oplus {H}^{0}(*)$, where * is a one-pointed space).

This schema can be used for other spaces like the torus (which can be also calculated from Kunneth’s formula).

Title | example of cohomology and Mayer-Vietoris sequence |
---|---|

Canonical name | ExampleOfCohomologyAndMayerVietorisSequence |

Date of creation | 2013-03-22 19:13:27 |

Last modified on | 2013-03-22 19:13:27 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 7 |

Author | joking (16130) |

Entry type | Example |

Classification | msc 55N33 |